[1]} (see also [2]}) observed that, in the presence of constant vorticity, the water wave system in \((\eta , \xi )\)  can still be expressed as a Hamiltonian system but in non-canonical form, namely
\(\partial _t \begin{pmatrix}\eta \\ \xi \end{pmatrix} =\begin{pmatrix}0 & 1 \\ -1 & \gamma \partial _x^{-1}\end{pmatrix} \begin{pmatrix}\partial _\eta H \\ \partial _\xi H\end{pmatrix} \,.\) 
